Computing the ideal of relations

Given two projective spaces $\mathbb P^n$ and $\mathbb P^m$ together with $m+1$ global sections of the invertible sheaf $\mathcal O_{\mathbb P^n}(d)$ (e.g. $m+1$ homogeneous polynomials of degree $d$ in the variables $x_0,\cdots,x_n$, say $f_0,\cdots,f_m$), we know that there exists a unique morphism $[f_0,\cdots,f_m] : \mathbb P^n \to \mathbb P^m$. Assume the projective spaces are considered over a noetherian ring ; the morphisms to the base are both projective, hence proper, which means $[f_0,\cdots,f_m]$ is a proper morphism, hence has closed image.

Question : Does there exist an algorithm already implemented in Sage to find the homogeneous ideal of relations of the image of the map $[f_0,\cdots,f_m]$? I've been messing around for a few days now and it seems to only involve linear algebra, so in the case where the base is the spectrum of a field there should be an algorithm, I just don't know how efficient it is or if it's implemented at all. I would not mind if the algorithm was slow, I just want it to work in small cases (i.e. small degree and small number of polynomials)!

1 answer

The missing key word is elimination, the sage method is elimination_ideal:

There will be no (projective) Spec in this answer, hope, i am translating correctly
the question into the affine (homogenos) setting.
Let us fix a (ring $R$, or maybe first a) field $F$.
We consider the two polynomial rings
$$A=A_m=F[x_0,x_1,\dots,x_m]$$
and
$$B=B_n=F[y_0,y_1,\dots,y_n]$$
and a map from $A$ to $B$ given formally by
$$x_k\to f_k(y_0, y_1,\dots,y_n)\ .$$
Here $f_k$ is a homogenous polynomial of degree $d$.
So the map $f$ (say)
$$ A_m\to B_n$$
induces a map
$$ {\mathbb P}_F^m\leftarrow {\mathbb P}_F^n\ .$$
(And we will never see the projective spaces again.)
Let us now work in the ring
$$
C = F[\ x_0,x_1,\dots,x_m\ ;\ y_0,y_1,\dots,y_n\ ]\ /\ J
$$
where $J$ is the ideal generated by the weighted homogenous polynomials
$$ y_k - f_k(x_0,x_1,\dots,x_m)\ .$$
We want and only need now to eliminate the $x$-variables.

Code example:

We use instead of $x_0,x_1,x_2$ the variables a,b,c.

And instead of $y_0, y_1,y_2,y_3,y_4,y_5,y_6$ the variables
s,t,u,v,w,x,y,z.

We work over rationals and consider the map of degree $d=4$ corresponding to: